On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles
Discrete & Computational Geometry
Covering the plane with convex polygons
Discrete & Computational Geometry
Journal of Algorithms
Fast algorithms for direct enclosures and direct dominances
Journal of Algorithms
Applications of random sampling in computational geometry, II
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
Computational geometry: algorithms and applications
Computational geometry: algorithms and applications
SCG '85 Proceedings of the first annual symposium on Computational geometry
The discrepancy method: randomness and complexity
The discrepancy method: randomness and complexity
Online conflict-free coloring for intervals
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Conflict-Free Coloring of Points and Simple Regions in the Plane
Discrete & Computational Geometry
On the chromatic number of some geometric hypergraphs
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
The Minimum Independence Number of a Hasse Diagram
Combinatorics, Probability and Computing
Conflict-free colorings of shallow discs
Proceedings of the twenty-second annual symposium on Computational geometry
Conflict-free coloring for rectangle ranges using O(n.382) colors
Proceedings of the nineteenth annual ACM symposium on Parallel algorithms and architectures
CJCDGCGT'05 Proceedings of the 7th China-Japan conference on Discrete geometry, combinatorics and graph theory
Conflict-Free colorings of rectangles ranges
STACS'06 Proceedings of the 23rd Annual conference on Theoretical Aspects of Computer Science
Conflict-Free coloring made stronger
SWAT'10 Proceedings of the 12th Scandinavian conference on Algorithm Theory
Coloring half-planes and bottomless rectangles
Computational Geometry: Theory and Applications
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Given a point set P in the plane, the Delaunay graph with respect to axis-parallel rectangles is a graph defined on the vertex set P, whose two points p,q ∈ P are connected by an edge if and only if there is a rectangle parallel to the coordinate axes that contains p and q, but no other elements of P. The following question of Even et al. [ELRS03] was motivated by a frequency assignment problem in cellular telephone networks. Does there exist a constant c 0 such that the Delaunay graph of any set of n points in general position in the plane contains an independent set of size at least cn? We answer this question in the negative, by proving that the largest independent set in a randomly and uniformly selected point set in the unit square is O(n log2 log n/log n), with probability tending to 1. We also show that our bound is not far from optimal, as the Delaunay graph of a uniform random set of n points almost surely has an independent set of size at least cn/ log n. We give two further applications of our methods. 1. We construct 2-dimensional n-element partially ordered sets such that the size of the largest independent sets of vertices in their Hasse diagrams is o(n). This answers a question of Matoušek and Přívětivý [MaP06] and improves a result of Kříž and Nešetřil [KrN91]. 2. For any positive integers c and d, we prove the existence of a planar point set with the property that no matter how we color its elements by c colors, we find an axis-parallel rectangle containing at least d points, all of which have the same color. This solves an old problem from [BrMP05].